Subsection RO Row Operations

An augmented matrix for a system of equations will save us the tedium of continually writing down the names of the variables as we solve the system. It will also release us from any dependence on the actual names of the variables. We have seen how certain operations we can perform on equations (Definition EO) will preserve their solutions (Theorem EOPSS). The next two definitions and the following theorem carry over these ideas to augmented matrices.

Definition RO Row Operations

The following three operations will transform an $m\times n$ matrix into a different matrix of the same size, and each is known as a row operation.

  1. Swap the locations of two rows.
  2. Multiply each entry of a single row by a nonzero quantity.
  3. Multiply each entry of one row by some quantity, and add these values to the entries in the same columns of a second row. Leave the first row the same after this operation, but replace the second row by the new values.
We will use a symbolic shorthand to describe these row operations:
  1. $\rowopswap{i}{j}$: Swap the location of rows $i$ and $j$.
  2. $\rowopmult{\alpha}{i}$: Multiply row $i$ by the nonzero scalar $\alpha$.
  3. $\rowopadd{\alpha}{i}{j}$: Multiply row $i$ by the scalar $\alpha$ and add to row $j$.

Definition REM Row-Equivalent Matrices

Two matrices, $A$ and $B$, are row-equivalent if one can be obtained from the other by a sequence of row operations.

Example TREM Two row-equivalent matrices

Notice that each of the three row operations is reversible (Exercise RREF.T10), so we do not have to be careful about the distinction between “$A$ is row-equivalent to $B$” and “$B$ is row-equivalent to $A$.” (Exercise RREF.T11)

The preceding definitions are designed to make the following theorem possible. It says that row-equivalent matrices represent systems of linear equations that have identical solution sets.

Theorem REMES Row-Equivalent Matrices represent Equivalent Systems

Suppose that $A$ and $B$ are row-equivalent augmented matrices. Then the systems of linear equations that they represent are equivalent systems.

Proof

So at this point, our strategy is to begin with a system of equations, represent the system by an augmented matrix, perform row operations (which will preserve solutions for the system) to get a “simpler” augmented matrix, convert back to a “simpler” system of equations and then solve that system, knowing that its solutions are those of the original system. Here is a rehash of Example US as an exercise in using our new tools.

Example USR Three equations, one solution, reprised
Sage RO Row Operations
(in context)